Concave and other generalizations of stochastic Gronwall inequalities

Abstract

We provide nonlinear generalizations of a class of stochastic Gronwall inequalities that have been studied by von Renesse and Scheutzow (2010), Scheutzow (2013), Xie and Zhang (2020) and Mehri and Scheutzow (2021). This class of stochastic Gronwall inequalities is a useful tool for SDEs.More precisely, we study generalizations of the Bihari-LaSalle type. Whilst in a closely connected article by the author convex generalizations are studied, we investigate here concave and other generalizations. These types of estimates are useful to obtain existence and uniqueness of global solutions of path-dependent SDEs driven by Lévy processes under one-sided non-Lipschitz monotonicity and coercivity assumptions.

Stability estimates for singular SDEs and applications

“…The proof of the next result shares some clear analogies with those of stochastic Grönwall lemmas, cf. [19,Lemma 3.8], as well as [9] for a general overview; however, in order not to impose any restriction on the 𝑚-th moment we want to estimate, we will exploit crucially the structure of the SDE which allows us to "recenter" it at each step, a property which doesn't hold in the more general setting of those results.…”

A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations

“…Hudde et al [13] extended this to the case p ∈ (1, ∞). For related stochastic Gronwall lemmas see, e.g., [29,32,16,21,31,9,10].…”